Effective theories for incompressible magnetoelastic shallow shells

TL;DR

This paper analyzes the asymptotic behavior of thin incompressible magnetoelastic shallow shells using $ ext{Gamma}$-convergence, combining variational, geometric, and topological methods to characterize their effective theories.

Contribution

It extends previous work on magnetoelastic plates to shallow shells, providing a rigorous asymptotic analysis and effective models for these structures.

Findings

Characterization of asymptotic behavior via $ ext{Gamma}$-convergence.

Development of compactness results up to rigid motions.

Derivation of effective theories for shallow shells.

Abstract

We characterize the asymptotic behaviour, in the sense of $\Gamma$-convergence, of a thin magnetoelastic shallow shell. The compactness is achieved up to rigid motions. For deformations, it relies on an approximation by rigid movements, whereas for magnetizations it is based on a careful consideration of the geometry of the deformed domain. The result is obtained by a combination of variational methods ($\Gamma$-convergence) with degree theory, fixed-point and geometrical arguments. The proof strategy relies on an adaptation of an analogous result for incompressible magnetoelastic plates from M. Bresciani in arXiv:2007.14122 and an application of results by I.Velcic on elastic shallow shells in arXiv:1102.2647.